When \(k=1\) , it is relatively simple to obtain the lower bound (REF ) by analyzing \(\mathrm {Im\,}\widetilde{m}_{\varepsilon }\)  and using (REF ). Indeed, this was done in [1]}. For larger \(k\) , however, \(\mathrm {Im\,}\widetilde{m}_{\varepsilon }\)  is given by a summation of \(\sim k/2\)  terms with alternating signs (see (REF )). Furthermore, it can be shown that each of those yields a Fourier multiplier whose \(L^p-L^q\)  norm is \(\gtrsim {\varepsilon }^{-\frac{d-1}{q} +\frac{d}{2}-k}\) . In other words, there is no `leading term' in the alternating sum (REF ). This makes it difficult to determine the lower bound for \(\Vert \mathrm {Im\,}\widetilde{m}_{\varepsilon }(D)\Vert _{p\rightarrow q}\) .
